For some definitions and notation, we refer to the monograph [1]}.
Given an ordered \(k\) -subset \(X\)  of \([0, 1]\) , let \(U[X]\)  denote the step-function on \([0,1]^2\)  with uniform steps of length \(1/k\)  in each variable, and values given by \((U(X_i,X_j))_{i\ne j\in [k]}\) , and 0 on the main diagonal (this latter is in order to avoid having to use values concentrated on a null-set). Throughout this paper we shall assume that \(X\)  is obtained by sampling \(k\)  independent and identically distributed points uniformly from \([0,1]\) , and then rearranging them in increasing order.
